aboutsummaryrefslogblamecommitdiff
path: root/frozen_deps/Cryptodome/PublicKey/ECC.py
blob: e83664bdea5011add4eb22e58964390435d73578 (plain) (tree)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































                                                                                                                                                      
# ===================================================================
#
# Copyright (c) 2015, Legrandin <[email protected]>
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
#    notice, this list of conditions and the following disclaimer.
# 2. Redistributions in binary form must reproduce the above copyright
#    notice, this list of conditions and the following disclaimer in
#    the documentation and/or other materials provided with the
#    distribution.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
# COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
# BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
# ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.
# ===================================================================

from __future__ import print_function

import re
import sys
import struct
import binascii
from collections import namedtuple

from Cryptodome.Util.py3compat import bord, tobytes, tostr, bchr, is_string
from Cryptodome.Util.number import bytes_to_long, long_to_bytes

from Cryptodome.Math.Numbers import Integer
from Cryptodome.Util.asn1 import (DerObjectId, DerOctetString, DerSequence,
                              DerBitString)

from Cryptodome.PublicKey import (_expand_subject_public_key_info,
                              _create_subject_public_key_info,
                              _extract_subject_public_key_info)

from Cryptodome.Util._raw_api import (load_pycryptodome_raw_lib, VoidPointer,
                                  SmartPointer, c_size_t, c_uint8_ptr,
                                  c_ulonglong)

from Cryptodome.Random import get_random_bytes
from Cryptodome.Random.random import getrandbits


_ec_lib = load_pycryptodome_raw_lib("Cryptodome.PublicKey._ec_ws", """
typedef void EcContext;
typedef void EcPoint;
int ec_ws_new_context(EcContext **pec_ctx,
                      const uint8_t *modulus,
                      const uint8_t *b,
                      const uint8_t *order,
                      size_t len,
                      uint64_t seed);
void ec_free_context(EcContext *ec_ctx);
int ec_ws_new_point(EcPoint **pecp,
                    const uint8_t *x,
                    const uint8_t *y,
                    size_t len,
                    const EcContext *ec_ctx);
void ec_free_point(EcPoint *ecp);
int ec_ws_get_xy(uint8_t *x,
                 uint8_t *y,
                 size_t len,
                 const EcPoint *ecp);
int ec_ws_double(EcPoint *p);
int ec_ws_add(EcPoint *ecpa, EcPoint *ecpb);
int ec_ws_scalar(EcPoint *ecp,
                 const uint8_t *k,
                 size_t len,
                 uint64_t seed);
int ec_ws_clone(EcPoint **pecp2, const EcPoint *ecp);
int ec_ws_copy(EcPoint *ecp1, const EcPoint *ecp2);
int ec_ws_cmp(const EcPoint *ecp1, const EcPoint *ecp2);
int ec_ws_neg(EcPoint *p);
int ec_ws_normalize(EcPoint *ecp);
int ec_ws_is_pai(EcPoint *ecp);
""")

_Curve = namedtuple("_Curve", "p b order Gx Gy G modulus_bits oid context desc openssh")
_curves = {}


p256_names = ["p256", "NIST P-256", "P-256", "prime256v1", "secp256r1",
              "nistp256"]


def init_p256():
    p = 0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff
    b = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b
    order = 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551
    Gx = 0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296
    Gy = 0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5

    p256_modulus = long_to_bytes(p, 32)
    p256_b = long_to_bytes(b, 32)
    p256_order = long_to_bytes(order, 32)

    ec_p256_context = VoidPointer()
    result = _ec_lib.ec_ws_new_context(ec_p256_context.address_of(),
                                       c_uint8_ptr(p256_modulus),
                                       c_uint8_ptr(p256_b),
                                       c_uint8_ptr(p256_order),
                                       c_size_t(len(p256_modulus)),
                                       c_ulonglong(getrandbits(64))
                                       )
    if result:
        raise ImportError("Error %d initializing P-256 context" % result)

    context = SmartPointer(ec_p256_context.get(), _ec_lib.ec_free_context)
    p256 = _Curve(Integer(p),
                  Integer(b),
                  Integer(order),
                  Integer(Gx),
                  Integer(Gy),
                  None,
                  256,
                  "1.2.840.10045.3.1.7",    # ANSI X9.62
                  context,
                  "NIST P-256",
                  "ecdsa-sha2-nistp256")
    global p256_names
    _curves.update(dict.fromkeys(p256_names, p256))


init_p256()
del init_p256


p384_names = ["p384", "NIST P-384", "P-384", "prime384v1", "secp384r1",
              "nistp384"]


def init_p384():
    p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000ffffffff
    b = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef
    order = 0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52973
    Gx = 0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760aB7
    Gy = 0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5F

    p384_modulus = long_to_bytes(p, 48)
    p384_b = long_to_bytes(b, 48)
    p384_order = long_to_bytes(order, 48)

    ec_p384_context = VoidPointer()
    result = _ec_lib.ec_ws_new_context(ec_p384_context.address_of(),
                                       c_uint8_ptr(p384_modulus),
                                       c_uint8_ptr(p384_b),
                                       c_uint8_ptr(p384_order),
                                       c_size_t(len(p384_modulus)),
                                       c_ulonglong(getrandbits(64))
                                       )
    if result:
        raise ImportError("Error %d initializing P-384 context" % result)

    context = SmartPointer(ec_p384_context.get(), _ec_lib.ec_free_context)
    p384 = _Curve(Integer(p),
                  Integer(b),
                  Integer(order),
                  Integer(Gx),
                  Integer(Gy),
                  None,
                  384,
                  "1.3.132.0.34",   # SEC 2
                  context,
                  "NIST P-384",
                  "ecdsa-sha2-nistp384")
    global p384_names
    _curves.update(dict.fromkeys(p384_names, p384))


init_p384()
del init_p384


p521_names = ["p521", "NIST P-521", "P-521", "prime521v1", "secp521r1",
              "nistp521"]


def init_p521():
    p = 0x000001ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
    b = 0x00000051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00
    order = 0x000001fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409
    Gx = 0x000000c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66
    Gy = 0x0000011839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650

    p521_modulus = long_to_bytes(p, 66)
    p521_b = long_to_bytes(b, 66)
    p521_order = long_to_bytes(order, 66)

    ec_p521_context = VoidPointer()
    result = _ec_lib.ec_ws_new_context(ec_p521_context.address_of(),
                                       c_uint8_ptr(p521_modulus),
                                       c_uint8_ptr(p521_b),
                                       c_uint8_ptr(p521_order),
                                       c_size_t(len(p521_modulus)),
                                       c_ulonglong(getrandbits(64))
                                       )
    if result:
        raise ImportError("Error %d initializing P-521 context" % result)

    context = SmartPointer(ec_p521_context.get(), _ec_lib.ec_free_context)
    p521 = _Curve(Integer(p),
                  Integer(b),
                  Integer(order),
                  Integer(Gx),
                  Integer(Gy),
                  None,
                  521,
                  "1.3.132.0.35",   # SEC 2
                  context,
                  "NIST P-521",
                  "ecdsa-sha2-nistp521")
    global p521_names
    _curves.update(dict.fromkeys(p521_names, p521))


init_p521()
del init_p521


class UnsupportedEccFeature(ValueError):
    pass


class EccPoint(object):
    """A class to abstract a point over an Elliptic Curve.

    The class support special methods for:

    * Adding two points: ``R = S + T``
    * In-place addition: ``S += T``
    * Negating a point: ``R = -T``
    * Comparing two points: ``if S == T: ...``
    * Multiplying a point by a scalar: ``R = S*k``
    * In-place multiplication by a scalar: ``T *= k``

    :ivar x: The affine X-coordinate of the ECC point
    :vartype x: integer

    :ivar y: The affine Y-coordinate of the ECC point
    :vartype y: integer

    :ivar xy: The tuple with X- and Y- coordinates
    """

    def __init__(self, x, y, curve="p256"):

        try:
            self._curve = _curves[curve]
        except KeyError:
            raise ValueError("Unknown curve name %s" % str(curve))
        self._curve_name = curve

        modulus_bytes = self.size_in_bytes()
        context = self._curve.context

        xb = long_to_bytes(x, modulus_bytes)
        yb = long_to_bytes(y, modulus_bytes)
        if len(xb) != modulus_bytes or len(yb) != modulus_bytes:
            raise ValueError("Incorrect coordinate length")

        self._point = VoidPointer()
        result = _ec_lib.ec_ws_new_point(self._point.address_of(),
                                         c_uint8_ptr(xb),
                                         c_uint8_ptr(yb),
                                         c_size_t(modulus_bytes),
                                         context.get())
        if result:
            if result == 15:
                raise ValueError("The EC point does not belong to the curve")
            raise ValueError("Error %d while instantiating an EC point" % result)

        # Ensure that object disposal of this Python object will (eventually)
        # free the memory allocated by the raw library for the EC point
        self._point = SmartPointer(self._point.get(),
                                   _ec_lib.ec_free_point)

    def set(self, point):
        self._point = VoidPointer()
        result = _ec_lib.ec_ws_clone(self._point.address_of(),
                                     point._point.get())
        if result:
            raise ValueError("Error %d while cloning an EC point" % result)

        self._point = SmartPointer(self._point.get(),
                                   _ec_lib.ec_free_point)
        return self

    def __eq__(self, point):
        return 0 == _ec_lib.ec_ws_cmp(self._point.get(), point._point.get())

    def __neg__(self):
        np = self.copy()
        result = _ec_lib.ec_ws_neg(np._point.get())
        if result:
            raise ValueError("Error %d while inverting an EC point" % result)
        return np

    def copy(self):
        """Return a copy of this point."""
        x, y = self.xy
        np = EccPoint(x, y, self._curve_name)
        return np

    def is_point_at_infinity(self):
        """``True`` if this is the point-at-infinity."""
        return self.xy == (0, 0)

    def point_at_infinity(self):
        """Return the point-at-infinity for the curve this point is on."""
        return EccPoint(0, 0, self._curve_name)

    @property
    def x(self):
        return self.xy[0]

    @property
    def y(self):
        return self.xy[1]

    @property
    def xy(self):
        modulus_bytes = self.size_in_bytes()
        xb = bytearray(modulus_bytes)
        yb = bytearray(modulus_bytes)
        result = _ec_lib.ec_ws_get_xy(c_uint8_ptr(xb),
                                      c_uint8_ptr(yb),
                                      c_size_t(modulus_bytes),
                                      self._point.get())
        if result:
            raise ValueError("Error %d while encoding an EC point" % result)

        return (Integer(bytes_to_long(xb)), Integer(bytes_to_long(yb)))

    def size_in_bytes(self):
        """Size of each coordinate, in bytes."""
        return (self.size_in_bits() + 7) // 8

    def size_in_bits(self):
        """Size of each coordinate, in bits."""
        return self._curve.modulus_bits

    def double(self):
        """Double this point (in-place operation).

        :Return:
            :class:`EccPoint` : this same object (to enable chaining)
        """

        result = _ec_lib.ec_ws_double(self._point.get())
        if result:
            raise ValueError("Error %d while doubling an EC point" % result)
        return self

    def __iadd__(self, point):
        """Add a second point to this one"""

        result = _ec_lib.ec_ws_add(self._point.get(), point._point.get())
        if result:
            if result == 16:
                raise ValueError("EC points are not on the same curve")
            raise ValueError("Error %d while adding two EC points" % result)
        return self

    def __add__(self, point):
        """Return a new point, the addition of this one and another"""

        np = self.copy()
        np += point
        return np

    def __imul__(self, scalar):
        """Multiply this point by a scalar"""

        if scalar < 0:
            raise ValueError("Scalar multiplication is only defined for non-negative integers")
        sb = long_to_bytes(scalar)
        result = _ec_lib.ec_ws_scalar(self._point.get(),
                                      c_uint8_ptr(sb),
                                      c_size_t(len(sb)),
                                      c_ulonglong(getrandbits(64)))
        if result:
            raise ValueError("Error %d during scalar multiplication" % result)
        return self

    def __mul__(self, scalar):
        """Return a new point, the scalar product of this one"""

        np = self.copy()
        np *= scalar
        return np

    def __rmul__(self, left_hand):
        return self.__mul__(left_hand)


# Last piece of initialization
p256_G = EccPoint(_curves['p256'].Gx, _curves['p256'].Gy, "p256")
p256 = _curves['p256']._replace(G=p256_G)
_curves.update(dict.fromkeys(p256_names, p256))
del p256_G, p256, p256_names

p384_G = EccPoint(_curves['p384'].Gx, _curves['p384'].Gy, "p384")
p384 = _curves['p384']._replace(G=p384_G)
_curves.update(dict.fromkeys(p384_names, p384))
del p384_G, p384, p384_names

p521_G = EccPoint(_curves['p521'].Gx, _curves['p521'].Gy, "p521")
p521 = _curves['p521']._replace(G=p521_G)
_curves.update(dict.fromkeys(p521_names, p521))
del p521_G, p521, p521_names


class EccKey(object):
    r"""Class defining an ECC key.
    Do not instantiate directly.
    Use :func:`generate`, :func:`construct` or :func:`import_key` instead.

    :ivar curve: The name of the ECC as defined in :numref:`curve_names`.
    :vartype curve: string

    :ivar pointQ: an ECC point representating the public component
    :vartype pointQ: :class:`EccPoint`

    :ivar d: A scalar representating the private component
    :vartype d: integer
    """

    def __init__(self, **kwargs):
        """Create a new ECC key

        Keywords:
          curve : string
            It must be *"p256"*, *"P-256"*, *"prime256v1"* or *"secp256r1"*.
          d : integer
            Only for a private key. It must be in the range ``[1..order-1]``.
          point : EccPoint
            Mandatory for a public key. If provided for a private key,
            the implementation will NOT check whether it matches ``d``.
        """

        kwargs_ = dict(kwargs)
        curve_name = kwargs_.pop("curve", None)
        self._d = kwargs_.pop("d", None)
        self._point = kwargs_.pop("point", None)
        if kwargs_:
            raise TypeError("Unknown parameters: " + str(kwargs_))

        if curve_name not in _curves:
            raise ValueError("Unsupported curve (%s)", curve_name)
        self._curve = _curves[curve_name]

        if self._d is None:
            if self._point is None:
                raise ValueError("Either private or public ECC component must be specified, not both")
        else:
            self._d = Integer(self._d)
            if not 1 <= self._d < self._curve.order:
                raise ValueError("Invalid ECC private component")

        self.curve = self._curve.desc

    def __eq__(self, other):
        if other.has_private() != self.has_private():
            return False

        return other.pointQ == self.pointQ

    def __repr__(self):
        if self.has_private():
            extra = ", d=%d" % int(self._d)
        else:
            extra = ""
        x, y = self.pointQ.xy
        return "EccKey(curve='%s', point_x=%d, point_y=%d%s)" % (self._curve.desc, x, y, extra)

    def has_private(self):
        """``True`` if this key can be used for making signatures or decrypting data."""

        return self._d is not None

    def _sign(self, z, k):
        assert 0 < k < self._curve.order

        order = self._curve.order
        blind = Integer.random_range(min_inclusive=1,
                                     max_exclusive=order)

        blind_d = self._d * blind
        inv_blind_k = (blind * k).inverse(order)

        r = (self._curve.G * k).x % order
        s = inv_blind_k * (blind * z + blind_d * r) % order
        return (r, s)

    def _verify(self, z, rs):
        order = self._curve.order
        sinv = rs[1].inverse(order)
        point1 = self._curve.G * ((sinv * z) % order)
        point2 = self.pointQ * ((sinv * rs[0]) % order)
        return (point1 + point2).x == rs[0]

    @property
    def d(self):
        if not self.has_private():
            raise ValueError("This is not a private ECC key")
        return self._d

    @property
    def pointQ(self):
        if self._point is None:
            self._point = self._curve.G * self._d
        return self._point

    def public_key(self):
        """A matching ECC public key.

        Returns:
            a new :class:`EccKey` object
        """

        return EccKey(curve=self._curve.desc, point=self.pointQ)

    def _export_subjectPublicKeyInfo(self, compress):

        # See 2.2 in RFC5480 and 2.3.3 in SEC1
        # The first byte is:
        # - 0x02:   compressed, only X-coordinate, Y-coordinate is even
        # - 0x03:   compressed, only X-coordinate, Y-coordinate is odd
        # - 0x04:   uncompressed, X-coordinate is followed by Y-coordinate
        #
        # PAI is in theory encoded as 0x00.

        modulus_bytes = self.pointQ.size_in_bytes()

        if compress:
            first_byte = 2 + self.pointQ.y.is_odd()
            public_key = (bchr(first_byte) +
                          self.pointQ.x.to_bytes(modulus_bytes))
        else:
            public_key = (b'\x04' +
                          self.pointQ.x.to_bytes(modulus_bytes) +
                          self.pointQ.y.to_bytes(modulus_bytes))

        unrestricted_oid = "1.2.840.10045.2.1"
        return _create_subject_public_key_info(unrestricted_oid,
                                               public_key,
                                               DerObjectId(self._curve.oid))

    def _export_private_der(self, include_ec_params=True):

        assert self.has_private()

        # ECPrivateKey ::= SEQUENCE {
        #           version        INTEGER { ecPrivkeyVer1(1) } (ecPrivkeyVer1),
        #           privateKey     OCTET STRING,
        #           parameters [0] ECParameters {{ NamedCurve }} OPTIONAL,
        #           publicKey  [1] BIT STRING OPTIONAL
        #    }

        # Public key - uncompressed form
        modulus_bytes = self.pointQ.size_in_bytes()
        public_key = (b'\x04' +
                      self.pointQ.x.to_bytes(modulus_bytes) +
                      self.pointQ.y.to_bytes(modulus_bytes))

        seq = [1,
               DerOctetString(self.d.to_bytes(modulus_bytes)),
               DerObjectId(self._curve.oid, explicit=0),
               DerBitString(public_key, explicit=1)]

        if not include_ec_params:
            del seq[2]

        return DerSequence(seq).encode()

    def _export_pkcs8(self, **kwargs):
        from Cryptodome.IO import PKCS8

        if kwargs.get('passphrase', None) is not None and 'protection' not in kwargs:
            raise ValueError("At least the 'protection' parameter should be present")

        unrestricted_oid = "1.2.840.10045.2.1"
        private_key = self._export_private_der(include_ec_params=False)
        result = PKCS8.wrap(private_key,
                            unrestricted_oid,
                            key_params=DerObjectId(self._curve.oid),
                            **kwargs)
        return result

    def _export_public_pem(self, compress):
        from Cryptodome.IO import PEM

        encoded_der = self._export_subjectPublicKeyInfo(compress)
        return PEM.encode(encoded_der, "PUBLIC KEY")

    def _export_private_pem(self, passphrase, **kwargs):
        from Cryptodome.IO import PEM

        encoded_der = self._export_private_der()
        return PEM.encode(encoded_der, "EC PRIVATE KEY", passphrase, **kwargs)

    def _export_private_clear_pkcs8_in_clear_pem(self):
        from Cryptodome.IO import PEM

        encoded_der = self._export_pkcs8()
        return PEM.encode(encoded_der, "PRIVATE KEY")

    def _export_private_encrypted_pkcs8_in_clear_pem(self, passphrase, **kwargs):
        from Cryptodome.IO import PEM

        assert passphrase
        if 'protection' not in kwargs:
            raise ValueError("At least the 'protection' parameter should be present")
        encoded_der = self._export_pkcs8(passphrase=passphrase, **kwargs)
        return PEM.encode(encoded_der, "ENCRYPTED PRIVATE KEY")

    def _export_openssh(self, compress):
        if self.has_private():
            raise ValueError("Cannot export OpenSSH private keys")

        desc = self._curve.openssh
        modulus_bytes = self.pointQ.size_in_bytes()

        if compress:
            first_byte = 2 + self.pointQ.y.is_odd()
            public_key = (bchr(first_byte) +
                          self.pointQ.x.to_bytes(modulus_bytes))
        else:
            public_key = (b'\x04' +
                          self.pointQ.x.to_bytes(modulus_bytes) +
                          self.pointQ.y.to_bytes(modulus_bytes))

        middle = desc.split("-")[2]
        comps = (tobytes(desc), tobytes(middle), public_key)
        blob = b"".join([struct.pack(">I", len(x)) + x for x in comps])
        return desc + " " + tostr(binascii.b2a_base64(blob))

    def export_key(self, **kwargs):
        """Export this ECC key.

        Args:
          format (string):
            The format to use for encoding the key:

            - ``'DER'``. The key will be encoded in ASN.1 DER format (binary).
              For a public key, the ASN.1 ``subjectPublicKeyInfo`` structure
              defined in `RFC5480`_ will be used.
              For a private key, the ASN.1 ``ECPrivateKey`` structure defined
              in `RFC5915`_ is used instead (possibly within a PKCS#8 envelope,
              see the ``use_pkcs8`` flag below).
            - ``'PEM'``. The key will be encoded in a PEM_ envelope (ASCII).
            - ``'OpenSSH'``. The key will be encoded in the OpenSSH_ format
              (ASCII, public keys only).

          passphrase (byte string or string):
            The passphrase to use for protecting the private key.

          use_pkcs8 (boolean):
            Only relevant for private keys.

            If ``True`` (default and recommended), the `PKCS#8`_ representation
            will be used.

            If ``False``, the much weaker `PEM encryption`_ mechanism will be used.

          protection (string):
            When a private key is exported with password-protection
            and PKCS#8 (both ``DER`` and ``PEM`` formats), this parameter MUST be
            present and be a valid algorithm supported by :mod:`Cryptodome.IO.PKCS8`.
            It is recommended to use ``PBKDF2WithHMAC-SHA1AndAES128-CBC``.

          compress (boolean):
            If ``True``, a more compact representation of the public key
            with the X-coordinate only is used.

            If ``False`` (default), the full public key will be exported.

        .. warning::
            If you don't provide a passphrase, the private key will be
            exported in the clear!

        .. note::
            When exporting a private key with password-protection and `PKCS#8`_
            (both ``DER`` and ``PEM`` formats), any extra parameters
            to ``export_key()`` will be passed to :mod:`Cryptodome.IO.PKCS8`.

        .. _PEM:        http://www.ietf.org/rfc/rfc1421.txt
        .. _`PEM encryption`: http://www.ietf.org/rfc/rfc1423.txt
        .. _`PKCS#8`:   http://www.ietf.org/rfc/rfc5208.txt
        .. _OpenSSH:    http://www.openssh.com/txt/rfc5656.txt
        .. _RFC5480:    https://tools.ietf.org/html/rfc5480
        .. _RFC5915:    http://www.ietf.org/rfc/rfc5915.txt

        Returns:
            A multi-line string (for PEM and OpenSSH) or bytes (for DER) with the encoded key.
        """

        args = kwargs.copy()
        ext_format = args.pop("format")
        if ext_format not in ("PEM", "DER", "OpenSSH"):
            raise ValueError("Unknown format '%s'" % ext_format)

        compress = args.pop("compress", False)

        if self.has_private():
            passphrase = args.pop("passphrase", None)
            if is_string(passphrase):
                passphrase = tobytes(passphrase)
                if not passphrase:
                    raise ValueError("Empty passphrase")
            use_pkcs8 = args.pop("use_pkcs8", True)
            if ext_format == "PEM":
                if use_pkcs8:
                    if passphrase:
                        return self._export_private_encrypted_pkcs8_in_clear_pem(passphrase, **args)
                    else:
                        return self._export_private_clear_pkcs8_in_clear_pem()
                else:
                    return self._export_private_pem(passphrase, **args)
            elif ext_format == "DER":
                # DER
                if passphrase and not use_pkcs8:
                    raise ValueError("Private keys can only be encrpyted with DER using PKCS#8")
                if use_pkcs8:
                    return self._export_pkcs8(passphrase=passphrase, **args)
                else:
                    return self._export_private_der()
            else:
                raise ValueError("Private keys cannot be exported in OpenSSH format")
        else:  # Public key
            if args:
                raise ValueError("Unexpected parameters: '%s'" % args)
            if ext_format == "PEM":
                return self._export_public_pem(compress)
            elif ext_format == "DER":
                return self._export_subjectPublicKeyInfo(compress)
            else:
                return self._export_openssh(compress)


def generate(**kwargs):
    """Generate a new private key on the given curve.

    Args:

      curve (string):
        Mandatory. It must be a curve name defined in :numref:`curve_names`.

      randfunc (callable):
        Optional. The RNG to read randomness from.
        If ``None``, :func:`Cryptodome.Random.get_random_bytes` is used.
    """

    curve_name = kwargs.pop("curve")
    curve = _curves[curve_name]
    randfunc = kwargs.pop("randfunc", get_random_bytes)
    if kwargs:
        raise TypeError("Unknown parameters: " + str(kwargs))

    d = Integer.random_range(min_inclusive=1,
                             max_exclusive=curve.order,
                             randfunc=randfunc)

    return EccKey(curve=curve_name, d=d)


def construct(**kwargs):
    """Build a new ECC key (private or public) starting
    from some base components.

    Args:

      curve (string):
        Mandatory. It must be a curve name defined in :numref:`curve_names`.

      d (integer):
        Only for a private key. It must be in the range ``[1..order-1]``.

      point_x (integer):
        Mandatory for a public key. X coordinate (affine) of the ECC point.

      point_y (integer):
        Mandatory for a public key. Y coordinate (affine) of the ECC point.

    Returns:
      :class:`EccKey` : a new ECC key object
    """

    curve_name = kwargs["curve"]
    curve = _curves[curve_name]
    point_x = kwargs.pop("point_x", None)
    point_y = kwargs.pop("point_y", None)

    if "point" in kwargs:
        raise TypeError("Unknown keyword: point")

    if None not in (point_x, point_y):
        # ValueError is raised if the point is not on the curve
        kwargs["point"] = EccPoint(point_x, point_y, curve_name)

    # Validate that the private key matches the public one
    d = kwargs.get("d", None)
    if d is not None and "point" in kwargs:
        pub_key = curve.G * d
        if pub_key.xy != (point_x, point_y):
            raise ValueError("Private and public ECC keys do not match")

    return EccKey(**kwargs)


def _import_public_der(curve_oid, ec_point):
    """Convert an encoded EC point into an EccKey object

    curve_name: string with the OID of the curve
    ec_point: byte string with the EC point (not DER encoded)

    """

    for curve_name, curve in _curves.items():
        if curve.oid == curve_oid:
            break
    else:
        raise UnsupportedEccFeature("Unsupported ECC curve (OID: %s)" % curve_oid)

    # See 2.2 in RFC5480 and 2.3.3 in SEC1
    # The first byte is:
    # - 0x02:   compressed, only X-coordinate, Y-coordinate is even
    # - 0x03:   compressed, only X-coordinate, Y-coordinate is odd
    # - 0x04:   uncompressed, X-coordinate is followed by Y-coordinate
    #
    # PAI is in theory encoded as 0x00.

    modulus_bytes = curve.p.size_in_bytes()
    point_type = bord(ec_point[0])

    # Uncompressed point
    if point_type == 0x04:
        if len(ec_point) != (1 + 2 * modulus_bytes):
            raise ValueError("Incorrect EC point length")
        x = Integer.from_bytes(ec_point[1:modulus_bytes+1])
        y = Integer.from_bytes(ec_point[modulus_bytes+1:])
    # Compressed point
    elif point_type in (0x02, 0x3):
        if len(ec_point) != (1 + modulus_bytes):
            raise ValueError("Incorrect EC point length")
        x = Integer.from_bytes(ec_point[1:])
        y = (x**3 - x*3 + curve.b).sqrt(curve.p)    # Short Weierstrass
        if point_type == 0x02 and y.is_odd():
            y = curve.p - y
        if point_type == 0x03 and y.is_even():
            y = curve.p - y
    else:
        raise ValueError("Incorrect EC point encoding")

    return construct(curve=curve_name, point_x=x, point_y=y)


def _import_subjectPublicKeyInfo(encoded, *kwargs):
    """Convert a subjectPublicKeyInfo into an EccKey object"""

    # See RFC5480

    # Parse the generic subjectPublicKeyInfo structure
    oid, ec_point, params = _expand_subject_public_key_info(encoded)

    # ec_point must be an encoded OCTET STRING
    # params is encoded ECParameters

    # We accept id-ecPublicKey, id-ecDH, id-ecMQV without making any
    # distiction for now.

    # Restrictions can be captured in the key usage certificate
    # extension
    unrestricted_oid = "1.2.840.10045.2.1"
    ecdh_oid = "1.3.132.1.12"
    ecmqv_oid = "1.3.132.1.13"

    if oid not in (unrestricted_oid, ecdh_oid, ecmqv_oid):
        raise UnsupportedEccFeature("Unsupported ECC purpose (OID: %s)" % oid)

    # Parameters are mandatory for all three types
    if not params:
        raise ValueError("Missing ECC parameters")

    # ECParameters ::= CHOICE {
    #   namedCurve         OBJECT IDENTIFIER
    #   -- implicitCurve   NULL
    #   -- specifiedCurve  SpecifiedECDomain
    # }
    #
    # implicitCurve and specifiedCurve are not supported (as per RFC)
    curve_oid = DerObjectId().decode(params).value

    return _import_public_der(curve_oid, ec_point)


def _import_private_der(encoded, passphrase, curve_oid=None):

    # See RFC5915 https://tools.ietf.org/html/rfc5915
    #
    # ECPrivateKey ::= SEQUENCE {
    #           version        INTEGER { ecPrivkeyVer1(1) } (ecPrivkeyVer1),
    #           privateKey     OCTET STRING,
    #           parameters [0] ECParameters {{ NamedCurve }} OPTIONAL,
    #           publicKey  [1] BIT STRING OPTIONAL
    #    }

    private_key = DerSequence().decode(encoded, nr_elements=(3, 4))
    if private_key[0] != 1:
        raise ValueError("Incorrect ECC private key version")

    try:
        parameters = DerObjectId(explicit=0).decode(private_key[2]).value
        if curve_oid is not None and parameters != curve_oid:
            raise ValueError("Curve mismatch")
        curve_oid = parameters
    except ValueError:
        pass

    if curve_oid is None:
        raise ValueError("No curve found")

    for curve_name, curve in _curves.items():
        if curve.oid == curve_oid:
            break
    else:
        raise UnsupportedEccFeature("Unsupported ECC curve (OID: %s)" % curve_oid)

    scalar_bytes = DerOctetString().decode(private_key[1]).payload
    modulus_bytes = curve.p.size_in_bytes()
    if len(scalar_bytes) != modulus_bytes:
        raise ValueError("Private key is too small")
    d = Integer.from_bytes(scalar_bytes)

    # Decode public key (if any)
    if len(private_key) == 4:
        public_key_enc = DerBitString(explicit=1).decode(private_key[3]).value
        public_key = _import_public_der(curve_oid, public_key_enc)
        point_x = public_key.pointQ.x
        point_y = public_key.pointQ.y
    else:
        point_x = point_y = None

    return construct(curve=curve_name, d=d, point_x=point_x, point_y=point_y)


def _import_pkcs8(encoded, passphrase):
    from Cryptodome.IO import PKCS8

    # From RFC5915, Section 1:
    #
    # Distributing an EC private key with PKCS#8 [RFC5208] involves including:
    # a) id-ecPublicKey, id-ecDH, or id-ecMQV (from [RFC5480]) with the
    #    namedCurve as the parameters in the privateKeyAlgorithm field; and
    # b) ECPrivateKey in the PrivateKey field, which is an OCTET STRING.

    algo_oid, private_key, params = PKCS8.unwrap(encoded, passphrase)

    # We accept id-ecPublicKey, id-ecDH, id-ecMQV without making any
    # distiction for now.
    unrestricted_oid = "1.2.840.10045.2.1"
    ecdh_oid = "1.3.132.1.12"
    ecmqv_oid = "1.3.132.1.13"

    if algo_oid not in (unrestricted_oid, ecdh_oid, ecmqv_oid):
        raise UnsupportedEccFeature("Unsupported ECC purpose (OID: %s)" % algo_oid)

    curve_oid = DerObjectId().decode(params).value

    return _import_private_der(private_key, passphrase, curve_oid)


def _import_x509_cert(encoded, *kwargs):

    sp_info = _extract_subject_public_key_info(encoded)
    return _import_subjectPublicKeyInfo(sp_info)


def _import_der(encoded, passphrase):

    try:
        return _import_subjectPublicKeyInfo(encoded, passphrase)
    except UnsupportedEccFeature as err:
        raise err
    except (ValueError, TypeError, IndexError):
        pass

    try:
        return _import_x509_cert(encoded, passphrase)
    except UnsupportedEccFeature as err:
        raise err
    except (ValueError, TypeError, IndexError):
        pass

    try:
        return _import_private_der(encoded, passphrase)
    except UnsupportedEccFeature as err:
        raise err
    except (ValueError, TypeError, IndexError):
        pass

    try:
        return _import_pkcs8(encoded, passphrase)
    except UnsupportedEccFeature as err:
        raise err
    except (ValueError, TypeError, IndexError):
        pass

    raise ValueError("Not an ECC DER key")


def _import_openssh_public(encoded):
    keystring = binascii.a2b_base64(encoded.split(b' ')[1])

    keyparts = []
    while len(keystring) > 4:
        lk = struct.unpack(">I", keystring[:4])[0]
        keyparts.append(keystring[4:4 + lk])
        keystring = keystring[4 + lk:]

    for curve_name, curve in _curves.items():
        middle = tobytes(curve.openssh.split("-")[2])
        if keyparts[1] == middle:
            break
    else:
        raise ValueError("Unsupported ECC curve")

    return _import_public_der(curve.oid, keyparts[2])


def _import_openssh_private_ecc(data, password):

    from ._openssh import (import_openssh_private_generic,
                           read_bytes, read_string, check_padding)

    ssh_name, decrypted = import_openssh_private_generic(data, password)

    name, decrypted = read_string(decrypted)
    if name not in _curves:
        raise UnsupportedEccFeature("Unsupported ECC curve %s" % name)
    curve = _curves[name]
    modulus_bytes = (curve.modulus_bits + 7) // 8

    public_key, decrypted = read_bytes(decrypted)

    if bord(public_key[0]) != 4:
        raise ValueError("Only uncompressed OpenSSH EC keys are supported")
    if len(public_key) != 2 * modulus_bytes + 1:
        raise ValueError("Incorrect public key length")

    point_x = Integer.from_bytes(public_key[1:1+modulus_bytes])
    point_y = Integer.from_bytes(public_key[1+modulus_bytes:])
    point = EccPoint(point_x, point_y, curve=name)

    private_key, decrypted = read_bytes(decrypted)
    d = Integer.from_bytes(private_key)

    _, padded = read_string(decrypted)  # Comment
    check_padding(padded)

    return EccKey(curve=name, d=d, point=point)


def import_key(encoded, passphrase=None):
    """Import an ECC key (public or private).

    Args:
      encoded (bytes or multi-line string):
        The ECC key to import.

        An ECC **public** key can be:

        - An X.509 certificate, binary (DER) or ASCII (PEM)
        - An X.509 ``subjectPublicKeyInfo``, binary (DER) or ASCII (PEM)
        - An OpenSSH line (e.g. the content of ``~/.ssh/id_ecdsa``, ASCII)

        An ECC **private** key can be:

        - In binary format (DER, see section 3 of `RFC5915`_ or `PKCS#8`_)
        - In ASCII format (PEM or `OpenSSH 6.5+`_)

        Private keys can be in the clear or password-protected.

        For details about the PEM encoding, see `RFC1421`_/`RFC1423`_.

      passphrase (byte string):
        The passphrase to use for decrypting a private key.
        Encryption may be applied protected at the PEM level or at the PKCS#8 level.
        This parameter is ignored if the key in input is not encrypted.

    Returns:
      :class:`EccKey` : a new ECC key object

    Raises:
      ValueError: when the given key cannot be parsed (possibly because
        the pass phrase is wrong).

    .. _RFC1421: http://www.ietf.org/rfc/rfc1421.txt
    .. _RFC1423: http://www.ietf.org/rfc/rfc1423.txt
    .. _RFC5915: http://www.ietf.org/rfc/rfc5915.txt
    .. _`PKCS#8`: http://www.ietf.org/rfc/rfc5208.txt
    .. _`OpenSSH 6.5+`: https://flak.tedunangst.com/post/new-openssh-key-format-and-bcrypt-pbkdf
    """

    from Cryptodome.IO import PEM

    encoded = tobytes(encoded)
    if passphrase is not None:
        passphrase = tobytes(passphrase)

    # PEM
    if encoded.startswith(b'-----BEGIN OPENSSH PRIVATE KEY'):
        text_encoded = tostr(encoded)
        openssh_encoded, marker, enc_flag = PEM.decode(text_encoded, passphrase)
        result = _import_openssh_private_ecc(openssh_encoded, passphrase)
        return result

    elif encoded.startswith(b'-----'):

        text_encoded = tostr(encoded)

        # Remove any EC PARAMETERS section
        # Ignore its content because the curve type must be already given in the key
        if sys.version_info[:2] != (2, 6):
            ecparams_start = "-----BEGIN EC PARAMETERS-----"
            ecparams_end = "-----END EC PARAMETERS-----"
            text_encoded = re.sub(ecparams_start + ".*?" + ecparams_end, "",
                                  text_encoded,
                                  flags=re.DOTALL)

        der_encoded, marker, enc_flag = PEM.decode(text_encoded, passphrase)
        if enc_flag:
            passphrase = None
        try:
            result = _import_der(der_encoded, passphrase)
        except UnsupportedEccFeature as uef:
            raise uef
        except ValueError:
            raise ValueError("Invalid DER encoding inside the PEM file")
        return result

    # OpenSSH
    if encoded.startswith(b'ecdsa-sha2-'):
        return _import_openssh_public(encoded)

    # DER
    if len(encoded) > 0 and bord(encoded[0]) == 0x30:
        return _import_der(encoded, passphrase)

    raise ValueError("ECC key format is not supported")


if __name__ == "__main__":

    import time

    d = 0xc51e4753afdec1e6b6c6a5b992f43f8dd0c7a8933072708b6522468b2ffb06fd

    point = _curves['p256'].G.copy()
    count = 3000

    start = time.time()
    for x in range(count):
        pointX = point * d
    print("(P-256 G)", (time.time() - start) / count * 1000, "ms")

    start = time.time()
    for x in range(count):
        pointX = pointX * d
    print("(P-256 arbitrary point)", (time.time() - start) / count * 1000, "ms")